simple function

In measure theory, a simple function is a function that is afinite linear combination

$$h=\sum _{k=1}^n{c}_k{\chi }_{A_k}$$

of characteristic functions, where the $c_k$ are real coefficients andevery $A_k$ is a measurable set with respect to a fixed measure space.

If the measure space is $ℝ$ and each $A_k$ is an interval,then the function is called a step function. Thus, every stepfunction is a simple function.

Simple functions are used in analysis to interpolate betweencharacteristic functions and measurable functions. In other words,characteristic functions are easy to integrate:

$${\int }_E{\chi }_A𝑑x=|A|,$$

while simple functions are not much harder to integrate:

$${\int }_E\sum _{k=1}^n{c}_k{\chi }_{A_k}dx=\sum _{k=1}^n{c}_k|A_k|.$$

To integrate a measurable function, one approximates it from below bysimple functions. Thus, simple functions can be used to define theLebesgue integral over a subset of the measure space.